On the Characters and the Plancherel Formula of Nilpotent Groups ...

SCATTERING THEORY WITH TWO HILBERT SPACES 351
Proof. Obvious from
THEOREM 6.2. Let W+ = W+( U, , U,; J) exist. W+ is a partial
isometry with initial projection PI if there is a unitary operator I on
liI to B2 which is (U, , +)-epuiwalent to J.
Proof. (6.1) is satisfied if J is replaced by J, which is permitted
because J is equivalent to J.
THEOREM 6.3. Let W+ = W+(U, , U,; J) exist. Assume that J*
(the adjoint of J) is a (U, , +)-asymptotic left inverse to J. Then W+
is semicomplete and partially isometric with initial projection PI . W+ is
complete if and only if W; = W+( U, , U,; J*) exists and J is a
(U, , +)-asymptotic left inverse to J*. If these conditions are satis$ed,
then w: = w;.
Proof. The first assertion follows because (6.1) is satisfied; in
fact
lim II J&(t) P&J 11’ = lim VU- t) J*Wdt> PIA PI+) = (PI+, PI+>
since J*JUI(t) PI - UI(t) PI . The second assertion follows from
Theorem 5.3. Theorem 5.3 also shows that W+Wi = Pz . Multiplica-
tion from the left with W$ then gives Wi = WY because
W$W+=P,, P,W;= WL,and W$P,= Wz.
7. APPLICATIONS TO UNIFORMLY PROPAGATIVE SYSTEMS
As the first application of the foregoing results, let us consider the
general wave equations discussed by Wilcox [S]. Here the spaces z$.
consist of complex m x 1 matrix-valued functions 4(x) on R”, with
the norms given by
i = L2, (7.1)
where E,(X) are positive-definite, m x m Hermitian matrices depend-
ing on x E Rn and where C+(X)* denotes the Hermitian conjugate of
I#(x). It is assumed that E,(X) = E1 is constant and that E,(X) is
uniformly bounded from above and from below. It follows that $r
and $a are the same vector space L? = (L2(P))m with different norms.